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        <title>Bases for Topologies</title>

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                <h1 id="title" titleSize="">
                    Bases for Topologies
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>Let $X$ be a set. To specify a <a href=https://zhaoshenzhai.github.io/mathwiki/topological_space.md class="internalLink references" title="Topological Space" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/topological_space.md&#34;, {&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/topological_space.md&#34;, {&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">topology</a> on $X$, one needs to give a complete description of its open sets. In practice, however, specifying a certain subcollection $\mc{B}\subseteq\pow(X)$ will suffice, In which case we say that the topology is <em>generated</em> by $\mc{B}$. Two questions arise:</p>
<ol>
<li>If $\mc{T}$ is a <em>given</em> topology on $X$, what subcollections $\mc{B}\subseteq\mc{T}$ generate $\mc{T}$?</li>
<li>What kinds of collections $\mc{B}\subseteq\pow(X)$ can specify <em>some</em> topology $\mc{T}$ on $X$?</li>
</ol>
<h2 id="generation-of-a-fixed-topology">Generation of a Fixed Topology</h2>
<p>Let $\mc{T}$ be a fixed topology on $X$.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>A collection $\mc{B}\subseteq\mc{T}$ is a <em>basis for $\mc{T}$</em> if every open set $U\in\mc{T}$ is a union of sets in $\mc{B}$.</div>

<p>Since all open sets $U\in\mc{T}$ are just unions of sets in $\mc{B}$, we call $\mc{B}$ the <em>basic open sets</em>.</p>
<br>
<p>  Equivalently, $\mc{B}\subseteq\mc{T}$ is a basis iff for each $x\in X$, its neighborhood <a href=https://zhaoshenzhai.github.io/mathwiki/filter.md class="internalLink references" title="Filter" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/filter.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/filter.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">filter</a> $\mc{N}_x$ (consisting of all sets $A\subseteq X$ such that $x\in U\subseteq A$ for some $U\in\mc{T}$) admits a <a href=https://zhaoshenzhai.github.io/mathwiki/filter.md/#base class="internalLink references" title="Filter" mathLink="" secID="base" secDisplay="filter base" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/filter.md/#base&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/filter.md/#base&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">filter base</a> in $\mc{B}$.</p>
<blockquote>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect hide">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">If $\mc{B}\subseteq\mc{T}$ is a basis and $x\in X$, then $\l\{V\in\mc{B}\st x\in V\r\}$ is a filter base for $\mc{N}_x$. Conversely, let $U$ be open. Every $x\in U$ admits a filter base $\mc{B}_x\subseteq\mc{B}$, giving us for each $x\in U$ some $B_x\in\mc{B}_x\subseteq\mc{B}$ with $x\in B_x\subseteq U$, so that $U=\bigcup_{x\in U}B_x$.<span style="float:right;">$\blacksquare$</span></span></div>

</blockquote>
<p>Thus, bases can be viewed as a global version of filter bases.</p>
<br>
<p>  To further thin down the subcollection and obtain $\mc{T}$ by closing under finite intersections as well, we make the following</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>A collection $\mc{S}\subseteq\mc{T}$ is a <em>subbasis for $\mc{T}$</em> if every open set $U\in\mc{T}$ is a union of finite intersections of sets in $\mc{S}$.</div>

<h2 id="generation-of-some-topology">Generation of Some Topology</h2>
<p>One way for a collection $\mc{B}\subseteq\pow(X)$ to generate <em>some</em> topology on $X$ is to let it satisfy some of the axioms for a topology on $X$, from which a topology can be obtained by closing it under the others.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>A collection $\mc{B}\subseteq\pow(X)$ is a <em>(saturated) basis</em> for a topology if $X$ and all pairwise intersections of sets in $\mc{B}$ are unions of sets in $\mc{B}$ (resp. in $\mc{B}$).</div>

<p>In other words, $\mc{B}$ is a basis if it covers $X$ and for every $B_1,B_2\in\mc{B}$ with $x\in B_1\cap B_2$, we have $x\in B\subseteq B_1\cap B_2$ for some $B\in\mc{B}$. In either case, we obtain a topology on $X$ by closing it under unions, i.e. by letting
$$\begin{equation}
\mc{T}_\mc{B}\coloneqq\l\{U\subseteq X\st\fa x\in U,\ex B\in\mc{B},x\in B\subseteq U\r\},
\end{equation}$$
in which case $\mc{B}$ is a basis for $\mc{T}_\mc{B}$.</p>
<blockquote>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect hide">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent"><p>Since every saturated basis is a basis, we prove the claim for when $\mc{B}$ is just a basis. That $\em\in\mc{T}_\mc{B}$ is vacuous, and $X\in\mc{T}_\mc{B}$ since $\mc{B}$ covers $X$. Closure under arbitrary unions is clear. If $U_1,U_2\in\mc{T}_\mc{B}$ and $x\in U_1\cap U_2$, then $x\in B_i\subseteq U_i$ for $i=1,2$ for some $B_1,B_2\in\mc{B}$, and so $x\in B\subseteq B_1\cap B_2\subseteq U_1\cap U_2$ for some $B\in\mc{B}$.</p>
<br>
<p>  Finally, it is clear that $\mc{B}$ is a basis for $\mc{T}_\mc{B}$ since every $U\in\mc{T}$ can be written as $U=\bigcup_{x\in U}B_x$, where each $B_x\in\mc{B}$ is such that $x\in B_x\subseteq U$.<span style="float:right;">$\blacksquare$</span></p>
</span></div>

</blockquote>
<p>One can be more vigorous by removing all the axioms of a topology and consider <em>any</em> collection $\mc{S}\subseteq\pow(X)$. This induces a basis $\mc{B}_\mc{S}$ consisting of finite intersections of sets in $\mc{S}$, from which we obtain a topology $\mc{T}_\mc{S}\coloneqq\mc{T}_{\mc{B}_\mc{S}}$ with $\mc{S}$ as a subbasis thereof.</p>
<blockquote>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect hide">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">We show that $\mc{B}_\mc{S}\coloneqq\l\{\bigcap_{i&lt;n}U_i\st U_i\in\mc{S}\r\}$ is a saturated basis. Indeed, that $X\in\mc{B}_\mc{S}$ follows from letting $n=0$. That $\mc{B}_\mc{S}$ is closed under pairwise intersections is clear, since $\bigcap_{i&lt;n}U_i\cap\bigcap_{i&lt;n&rsquo;}U&rsquo;_i$ is still a finite intersection.<span style="float:right;">$\blacksquare$</span></span></div>

</blockquote>
<h1 id="rigidity-of-bases">Rigidity of Bases</h1>
<p>Many topological properties of $X$ can be detected (or defined) by the existence of certain bases thereof.</p>
<h2 id="weight-and-character">Weight and Character</h2>
<p>To begin, we have the following basic notion.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>The <em>weight</em> of $X$ is the minimal <a href=https://zhaoshenzhai.github.io/mathwiki/cardinal.md class="internalLink references ghostLink" title="cardinality" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cardinal.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cardinal.md&#34;, &#34;nopPage&#34;);">cardinality</a> of a basis for $X$, denoted $w(X)$.</div>

<p>If $w(X)\leq\aleph_0$, then $X$ is said to be <em><a href=https://zhaoshenzhai.github.io/mathwiki/second_countable_space.md class="internalLink constructions" title="Second Countable Space" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/second_countable_space.md&#34;, {&#34;Date&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Second Countable Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/second_countable_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Second Countable Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/second_countable_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/second_countable_space.md&#34;, {&#34;Date&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Second Countable Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/second_countable_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">second countable</a></em>.</p>
<br>
<p>  A pointwise-version of the above is the <em>character</em> $\chi(x,X)$ of a point $x\in X$, defined as the minimal cardinality of a filter base for $\mc{N}_x$. Taking the supremum over all $x\in X$ gives the <em>character</em> $\chi(X)$ of $X$. If $\chi(X)\leq\aleph_0$, then $X$ is said to be <em><a href=https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md class="internalLink constructions ghostLink" title="first-countable" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md&#34;, &#34;nopPage&#34;);">first-countable</a></em>.</p>
<h2 id="local-properties">Local Properties</h2>


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                July 11, 2024 | Zhaoshen Zhai

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